Question:The number y is one-fourth of x less than the number x. Which equation represents the relationship between x and...

1. TRANSLATE the word phrase carefully

• The phrase "y is one-fourth of x less than the number x" breaks down as:
  • Start with "the number x"
  • Subtract "one-fourth of x" from it
  • This gives us y

• In mathematical notation: \(\mathrm{y = x - \frac{1}{4}x}\)

2. SIMPLIFY by combining like terms

• We have: \(\mathrm{y = x - \frac{1}{4}x}\)
• Rewrite x as \(\mathrm{\frac{4}{4}x}\): \(\mathrm{y = \frac{4}{4}x - \frac{1}{4}x}\)
• Combine the fractions: \(\mathrm{y = \frac{4-1}{4}x = \frac{3}{4}x}\)

Answer: D) \(\mathrm{y = \frac{3}{4}x}\)

Why Students Usually Falter on This Problem

Most Common Error Path:

Weak TRANSLATE skill: Students often misread "one-fourth of x less than x" as simply "one-fourth of x," ignoring the "less than x" part entirely.

They think: "y is one-fourth of x" → \(\mathrm{y = \frac{1}{4}x}\)

This leads them to select Choice B (\(\mathrm{y = \frac{1}{4}x}\)).

Second Most Common Error:

Poor TRANSLATE reasoning: Students interpret "one-fourth of x less than x" as "x minus one-fourth" instead of "x minus one-fourth of x."

They think: "less than x by one-fourth" → \(\mathrm{y = x - \frac{1}{4}}\)

This leads them to select Choice A (\(\mathrm{y = x - \frac{1}{4}}\)).

The Bottom Line:

The key challenge is carefully parsing the English phrase. "One-fourth of x less than x" means you start with x and subtract one-fourth OF x (not just subtract one-fourth). The word "of" is crucial—it means multiply, so "one-fourth of x" = \(\mathrm{\frac{1}{4}x}\).